You Do Need power plug adapter while traveling from Anguilla to Channel Islands (Guernsey & Jersey).

Here is Why You Need An Adapter

The outlet and plug type used in Channel Islands (Guernsey & Jersey) are C and G and in Anguilla are A and B. This is how they look like.

Anguilla

In Anguilla, outlets of type A and B are in use.

Type A

Type B

Channel Islands (Guernsey & Jersey)

In Channel Islands (Guernsey & Jersey), outlets of type C and G are in use.

Type C

Type G